Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [486,2,Mod(19,486)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(486, base_ring=CyclotomicField(54))
chi = DirichletCharacter(H, H._module([52]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("486.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 486 = 2 \cdot 3^{5} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 486.g (of order \(27\), degree \(18\), not minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(3.88072953823\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{27})\) |
Twist minimal: | no (minimal twist has level 162) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{27}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.993238 | + | 0.116093i | 0 | 0.973045 | − | 0.230616i | −2.13185 | − | 1.07065i | 0 | 0.413572 | + | 1.38143i | −0.939693 | + | 0.342020i | 0 | 2.24173 | + | 0.815922i | ||||||
19.2 | −0.993238 | + | 0.116093i | 0 | 0.973045 | − | 0.230616i | −0.907364 | − | 0.455695i | 0 | −0.755228 | − | 2.52264i | −0.939693 | + | 0.342020i | 0 | 0.954132 | + | 0.347276i | ||||||
19.3 | −0.993238 | + | 0.116093i | 0 | 0.973045 | − | 0.230616i | 0.110030 | + | 0.0552591i | 0 | −0.0823094 | − | 0.274932i | −0.939693 | + | 0.342020i | 0 | −0.115701 | − | 0.0421118i | ||||||
19.4 | −0.993238 | + | 0.116093i | 0 | 0.973045 | − | 0.230616i | 2.54814 | + | 1.27972i | 0 | 0.518331 | + | 1.73135i | −0.939693 | + | 0.342020i | 0 | −2.67948 | − | 0.975250i | ||||||
37.1 | −0.286803 | + | 0.957990i | 0 | −0.835488 | − | 0.549509i | −0.423867 | + | 0.982634i | 0 | −0.264903 | − | 4.54820i | 0.766044 | − | 0.642788i | 0 | −0.819786 | − | 0.687882i | ||||||
37.2 | −0.286803 | + | 0.957990i | 0 | −0.835488 | − | 0.549509i | −0.417669 | + | 0.968265i | 0 | −0.0919975 | − | 1.57954i | 0.766044 | − | 0.642788i | 0 | −0.807799 | − | 0.677824i | ||||||
37.3 | −0.286803 | + | 0.957990i | 0 | −0.835488 | − | 0.549509i | −0.266390 | + | 0.617563i | 0 | 0.0791338 | + | 1.35867i | 0.766044 | − | 0.642788i | 0 | −0.515217 | − | 0.432318i | ||||||
37.4 | −0.286803 | + | 0.957990i | 0 | −0.835488 | − | 0.549509i | 0.757906 | − | 1.75702i | 0 | 0.278553 | + | 4.78256i | 0.766044 | − | 0.642788i | 0 | 1.46584 | + | 1.22999i | ||||||
73.1 | 0.893633 | − | 0.448799i | 0 | 0.597159 | − | 0.802123i | −0.308002 | + | 1.02880i | 0 | −2.06499 | + | 4.78718i | 0.173648 | − | 0.984808i | 0 | 0.186483 | + | 1.05760i | ||||||
73.2 | 0.893633 | − | 0.448799i | 0 | 0.597159 | − | 0.802123i | −0.195520 | + | 0.653084i | 0 | 0.386069 | − | 0.895009i | 0.173648 | − | 0.984808i | 0 | 0.118380 | + | 0.671366i | ||||||
73.3 | 0.893633 | − | 0.448799i | 0 | 0.597159 | − | 0.802123i | 0.267154 | − | 0.892356i | 0 | 1.11188 | − | 2.57764i | 0.173648 | − | 0.984808i | 0 | −0.161751 | − | 0.917337i | ||||||
73.4 | 0.893633 | − | 0.448799i | 0 | 0.597159 | − | 0.802123i | 0.750365 | − | 2.50640i | 0 | 0.318489 | − | 0.738341i | 0.173648 | − | 0.984808i | 0 | −0.454317 | − | 2.57656i | ||||||
91.1 | −0.0581448 | − | 0.998308i | 0 | −0.993238 | + | 0.116093i | −3.64873 | − | 0.864766i | 0 | 1.03565 | − | 1.39113i | 0.173648 | + | 0.984808i | 0 | −0.651148 | + | 3.69284i | ||||||
91.2 | −0.0581448 | − | 0.998308i | 0 | −0.993238 | + | 0.116093i | −1.08715 | − | 0.257659i | 0 | 1.38530 | − | 1.86078i | 0.173648 | + | 0.984808i | 0 | −0.194011 | + | 1.10029i | ||||||
91.3 | −0.0581448 | − | 0.998308i | 0 | −0.993238 | + | 0.116093i | −0.789983 | − | 0.187229i | 0 | −1.95256 | + | 2.62274i | 0.173648 | + | 0.984808i | 0 | −0.140979 | + | 0.799532i | ||||||
91.4 | −0.0581448 | − | 0.998308i | 0 | −0.993238 | + | 0.116093i | 3.39069 | + | 0.803609i | 0 | −1.32018 | + | 1.77332i | 0.173648 | + | 0.984808i | 0 | 0.605098 | − | 3.43168i | ||||||
127.1 | −0.686242 | + | 0.727374i | 0 | −0.0581448 | − | 0.998308i | −1.83788 | − | 0.214818i | 0 | −0.0854199 | + | 0.0428995i | 0.766044 | + | 0.642788i | 0 | 1.41748 | − | 1.18941i | ||||||
127.2 | −0.686242 | + | 0.727374i | 0 | −0.0581448 | − | 0.998308i | −1.08261 | − | 0.126540i | 0 | −0.387485 | + | 0.194602i | 0.766044 | + | 0.642788i | 0 | 0.834977 | − | 0.700629i | ||||||
127.3 | −0.686242 | + | 0.727374i | 0 | −0.0581448 | − | 0.998308i | 1.97104 | + | 0.230382i | 0 | 0.868864 | − | 0.436360i | 0.766044 | + | 0.642788i | 0 | −1.52019 | + | 1.27559i | ||||||
127.4 | −0.686242 | + | 0.727374i | 0 | −0.0581448 | − | 0.998308i | 3.71787 | + | 0.434557i | 0 | −3.25050 | + | 1.63246i | 0.766044 | + | 0.642788i | 0 | −2.86744 | + | 2.40607i | ||||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
81.g | even | 27 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 486.2.g.a | 72 | |
3.b | odd | 2 | 1 | 162.2.g.a | ✓ | 72 | |
81.g | even | 27 | 1 | inner | 486.2.g.a | 72 | |
81.h | odd | 54 | 1 | 162.2.g.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
162.2.g.a | ✓ | 72 | 3.b | odd | 2 | 1 | |
162.2.g.a | ✓ | 72 | 81.h | odd | 54 | 1 | |
486.2.g.a | 72 | 1.a | even | 1 | 1 | trivial | |
486.2.g.a | 72 | 81.g | even | 27 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{72} - 9 T_{5}^{70} - 30 T_{5}^{69} + 63 T_{5}^{68} + 81 T_{5}^{67} - 1665 T_{5}^{66} + \cdots + 18487617876729 \) acting on \(S_{2}^{\mathrm{new}}(486, [\chi])\).